src/HOL/Proofs/Lambda/LambdaType.thy
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(*  Title:      HOL/Proofs/Lambda/LambdaType.thy
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    Author:     Stefan Berghofer
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    Copyright   2000 TU Muenchen
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*)
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header {* Simply-typed lambda terms *}
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theory LambdaType imports ListApplication begin
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subsection {* Environments *}
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definition
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  shift :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"  ("_<_:_>" [90, 0, 0] 91) where
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  "e<i:a> = (\<lambda>j. if j < i then e j else if j = i then a else e (j - 1))"
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notation (xsymbols)
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  shift  ("_\<langle>_:_\<rangle>" [90, 0, 0] 91)
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notation (HTML output)
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  shift  ("_\<langle>_:_\<rangle>" [90, 0, 0] 91)
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lemma shift_eq [simp]: "i = j \<Longrightarrow> (e\<langle>i:T\<rangle>) j = T"
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  by (simp add: shift_def)
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lemma shift_gt [simp]: "j < i \<Longrightarrow> (e\<langle>i:T\<rangle>) j = e j"
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  by (simp add: shift_def)
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lemma shift_lt [simp]: "i < j \<Longrightarrow> (e\<langle>i:T\<rangle>) j = e (j - 1)"
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  by (simp add: shift_def)
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lemma shift_commute [simp]: "e\<langle>i:U\<rangle>\<langle>0:T\<rangle> = e\<langle>0:T\<rangle>\<langle>Suc i:U\<rangle>"
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  by (rule ext) (simp_all add: shift_def split: nat.split)
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subsection {* Types and typing rules *}
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datatype type =
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    Atom nat
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  | Fun type type    (infixr "\<Rightarrow>" 200)
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inductive typing :: "(nat \<Rightarrow> type) \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool"  ("_ \<turnstile> _ : _" [50, 50, 50] 50)
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  where
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    Var [intro!]: "env x = T \<Longrightarrow> env \<turnstile> Var x : T"
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  | Abs [intro!]: "env\<langle>0:T\<rangle> \<turnstile> t : U \<Longrightarrow> env \<turnstile> Abs t : (T \<Rightarrow> U)"
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  | App [intro!]: "env \<turnstile> s : T \<Rightarrow> U \<Longrightarrow> env \<turnstile> t : T \<Longrightarrow> env \<turnstile> (s \<degree> t) : U"
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inductive_cases typing_elims [elim!]:
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  "e \<turnstile> Var i : T"
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  "e \<turnstile> t \<degree> u : T"
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  "e \<turnstile> Abs t : T"
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primrec
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  typings :: "(nat \<Rightarrow> type) \<Rightarrow> dB list \<Rightarrow> type list \<Rightarrow> bool"
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where
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    "typings e [] Ts = (Ts = [])"
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  | "typings e (t # ts) Ts =
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      (case Ts of
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        [] \<Rightarrow> False
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      | T # Ts \<Rightarrow> e \<turnstile> t : T \<and> typings e ts Ts)"
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abbreviation
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  typings_rel :: "(nat \<Rightarrow> type) \<Rightarrow> dB list \<Rightarrow> type list \<Rightarrow> bool"
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    ("_ ||- _ : _" [50, 50, 50] 50) where
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  "env ||- ts : Ts == typings env ts Ts"
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notation (latex)
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  typings_rel  ("_ \<tturnstile> _ : _" [50, 50, 50] 50)
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abbreviation
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  funs :: "type list \<Rightarrow> type \<Rightarrow> type"  (infixr "=>>" 200) where
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  "Ts =>> T == foldr Fun Ts T"
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notation (latex)
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  funs  (infixr "\<Rrightarrow>" 200)
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subsection {* Some examples *}
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schematic_lemma "e \<turnstile> Abs (Abs (Abs (Var 1 \<degree> (Var 2 \<degree> Var 1 \<degree> Var 0)))) : ?T"
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  by force
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schematic_lemma "e \<turnstile> Abs (Abs (Abs (Var 2 \<degree> Var 0 \<degree> (Var 1 \<degree> Var 0)))) : ?T"
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  by force
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subsection {* Lists of types *}
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lemma lists_typings:
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    "e \<tturnstile> ts : Ts \<Longrightarrow> listsp (\<lambda>t. \<exists>T. e \<turnstile> t : T) ts"
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  apply (induct ts arbitrary: Ts)
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   apply (case_tac Ts)
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     apply simp
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     apply (rule listsp.Nil)
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    apply simp
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  apply (case_tac Ts)
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   apply simp
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  apply simp
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  apply (rule listsp.Cons)
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   apply blast
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  apply blast
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  done
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lemma types_snoc: "e \<tturnstile> ts : Ts \<Longrightarrow> e \<turnstile> t : T \<Longrightarrow> e \<tturnstile> ts @ [t] : Ts @ [T]"
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  apply (induct ts arbitrary: Ts)
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  apply simp
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  apply (case_tac Ts)
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  apply simp+
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  done
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lemma types_snoc_eq: "e \<tturnstile> ts @ [t] : Ts @ [T] =
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  (e \<tturnstile> ts : Ts \<and> e \<turnstile> t : T)"
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  apply (induct ts arbitrary: Ts)
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  apply (case_tac Ts)
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  apply simp+
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  apply (case_tac Ts)
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  apply (case_tac "ts @ [t]")
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  apply simp+
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  done
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lemma rev_exhaust2 [extraction_expand]:
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  obtains (Nil) "xs = []"  |  (snoc) ys y where "xs = ys @ [y]"
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  -- {* Cannot use @{text rev_exhaust} from the @{text List}
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    theory, since it is not constructive *}
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  apply (subgoal_tac "\<forall>ys. xs = rev ys \<longrightarrow> thesis")
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  apply (erule_tac x="rev xs" in allE)
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  apply simp
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  apply (rule allI)
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  apply (rule impI)
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  apply (case_tac ys)
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  apply simp
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  apply simp
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  done
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lemma types_snocE: "e \<tturnstile> ts @ [t] : Ts \<Longrightarrow>
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  (\<And>Us U. Ts = Us @ [U] \<Longrightarrow> e \<tturnstile> ts : Us \<Longrightarrow> e \<turnstile> t : U \<Longrightarrow> P) \<Longrightarrow> P"
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  apply (cases Ts rule: rev_exhaust2)
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  apply simp
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  apply (case_tac "ts @ [t]")
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  apply (simp add: types_snoc_eq)+
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  done
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subsection {* n-ary function types *}
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lemma list_app_typeD:
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    "e \<turnstile> t \<degree>\<degree> ts : T \<Longrightarrow> \<exists>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<and> e \<tturnstile> ts : Ts"
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  apply (induct ts arbitrary: t T)
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   apply simp
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  apply (rename_tac a b t T)
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  apply atomize
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  apply simp
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  apply (erule_tac x = "t \<degree> a" in allE)
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  apply (erule_tac x = T in allE)
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  apply (erule impE)
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   apply assumption
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  apply (elim exE conjE)
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply (rule_tac x = "Ta # Ts" in exI)
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  apply simp
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  done
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lemma list_app_typeE:
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  "e \<turnstile> t \<degree>\<degree> ts : T \<Longrightarrow> (\<And>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> C) \<Longrightarrow> C"
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  by (insert list_app_typeD) fast
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lemma list_app_typeI:
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    "e \<turnstile> t : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> e \<turnstile> t \<degree>\<degree> ts : T"
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  apply (induct ts arbitrary: t T Ts)
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   apply simp
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  apply (rename_tac a b t T Ts)
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  apply atomize
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  apply (case_tac Ts)
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   apply simp
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  apply simp
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  apply (erule_tac x = "t \<degree> a" in allE)
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  apply (erule_tac x = T in allE)
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  apply (rename_tac list)
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  apply (erule_tac x = list in allE)
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  apply (erule impE)
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   apply (erule conjE)
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   apply (erule typing.App)
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   apply assumption
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  apply blast
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  done
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text {*
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For the specific case where the head of the term is a variable,
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the following theorems allow to infer the types of the arguments
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without analyzing the typing derivation. This is crucial
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for program extraction.
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*}
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theorem var_app_type_eq:
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  "e \<turnstile> Var i \<degree>\<degree> ts : T \<Longrightarrow> e \<turnstile> Var i \<degree>\<degree> ts : U \<Longrightarrow> T = U"
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  apply (induct ts arbitrary: T U rule: rev_induct)
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  apply simp
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  apply (ind_cases "e \<turnstile> Var i : T" for T)
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  apply (ind_cases "e \<turnstile> Var i : T" for T)
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  apply simp
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  apply simp
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply atomize
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  apply (erule_tac x="Ta \<Rightarrow> T" in allE)
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  apply (erule_tac x="Tb \<Rightarrow> U" in allE)
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  apply (erule impE)
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  apply assumption
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  apply (erule impE)
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  apply assumption
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  apply simp
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  done
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lemma var_app_types: "e \<turnstile> Var i \<degree>\<degree> ts \<degree>\<degree> us : T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow>
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  e \<turnstile> Var i \<degree>\<degree> ts : U \<Longrightarrow> \<exists>Us. U = Us \<Rrightarrow> T \<and> e \<tturnstile> us : Us"
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  apply (induct us arbitrary: ts Ts U)
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  apply simp
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  apply (erule var_app_type_eq)
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  apply assumption
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  apply simp
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  apply (rename_tac a b ts Ts U)
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  apply atomize
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  apply (case_tac U)
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  apply (rule FalseE)
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  apply simp
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  apply (erule list_app_typeE)
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply (drule_tac T="Atom nat" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
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  apply assumption
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  apply simp
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  apply (erule_tac x="ts @ [a]" in allE)
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  apply (erule_tac x="Ts @ [type1]" in allE)
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  apply (erule_tac x="type2" in allE)
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  apply simp
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  apply (erule impE)
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  apply (rule types_snoc)
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  apply assumption
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  apply (erule list_app_typeE)
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply (drule_tac T="type1 \<Rightarrow> type2" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
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  apply assumption
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   242
  apply simp
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  apply (erule impE)
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  apply (rule typing.App)
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  apply assumption
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  apply (erule list_app_typeE)
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply (frule_tac T="type1 \<Rightarrow> type2" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
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  apply assumption
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   250
  apply simp
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  apply (erule exE)
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  apply (rule_tac x="type1 # Us" in exI)
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  apply simp
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  apply (erule list_app_typeE)
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  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
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  apply (frule_tac T="type1 \<Rightarrow> Us \<Rrightarrow> T" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
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  apply assumption
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  apply simp
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  done
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lemma var_app_typesE: "e \<turnstile> Var i \<degree>\<degree> ts : T \<Longrightarrow>
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  (\<And>Ts. e \<turnstile> Var i : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> P) \<Longrightarrow> P"
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  apply (drule var_app_types [of _ _ "[]", simplified])
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  apply (iprover intro: typing.Var)+
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  done
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lemma abs_typeE: "e \<turnstile> Abs t : T \<Longrightarrow> (\<And>U V. e\<langle>0:U\<rangle> \<turnstile> t : V \<Longrightarrow> P) \<Longrightarrow> P"
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  apply (cases T)
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  apply (rule FalseE)
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  apply (erule typing.cases)
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  apply simp_all
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  apply atomize
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  apply (erule_tac x="type1" in allE)
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  apply (erule_tac x="type2" in allE)
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  apply (erule mp)
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  apply (erule typing.cases)
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  apply simp_all
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  done
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subsection {* Lifting preserves well-typedness *}
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lemma lift_type [intro!]: "e \<turnstile> t : T \<Longrightarrow> e\<langle>i:U\<rangle> \<turnstile> lift t i : T"
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  by (induct arbitrary: i U set: typing) auto
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lemma lift_types:
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  "e \<tturnstile> ts : Ts \<Longrightarrow> e\<langle>i:U\<rangle> \<tturnstile> (map (\<lambda>t. lift t i) ts) : Ts"
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  apply (induct ts arbitrary: Ts)
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   apply simp
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  apply (case_tac Ts)
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   apply auto
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  done
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subsection {* Substitution lemmas *}
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lemma subst_lemma:
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    "e \<turnstile> t : T \<Longrightarrow> e' \<turnstile> u : U \<Longrightarrow> e = e'\<langle>i:U\<rangle> \<Longrightarrow> e' \<turnstile> t[u/i] : T"
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  apply (induct arbitrary: e' i U u set: typing)
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    apply (rule_tac x = x and y = i in linorder_cases)
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      apply auto
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  apply blast
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  done
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lemma substs_lemma:
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  "e \<turnstile> u : T \<Longrightarrow> e\<langle>i:T\<rangle> \<tturnstile> ts : Ts \<Longrightarrow>
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   307
     e \<tturnstile> (map (\<lambda>t. t[u/i]) ts) : Ts"
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  apply (induct ts arbitrary: Ts)
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   apply (case_tac Ts)
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    apply simp
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   apply simp
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  apply atomize
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  apply (case_tac Ts)
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   apply simp
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  apply simp
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  apply (erule conjE)
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  apply (erule (1) subst_lemma)
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  apply (rule refl)
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  done
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subsection {* Subject reduction *}
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lemma subject_reduction: "e \<turnstile> t : T \<Longrightarrow> t \<rightarrow>\<^sub>\<beta> t' \<Longrightarrow> e \<turnstile> t' : T"
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  apply (induct arbitrary: t' set: typing)
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    apply blast
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   apply blast
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  apply atomize
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  apply (ind_cases "s \<degree> t \<rightarrow>\<^sub>\<beta> t'" for s t t')
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    apply hypsubst
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    apply (ind_cases "env \<turnstile> Abs t : T \<Rightarrow> U" for env t T U)
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    apply (rule subst_lemma)
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      apply assumption
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     apply assumption
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    apply (rule ext)
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    apply (case_tac x)
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     apply auto
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  done
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theorem subject_reduction': "t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> e \<turnstile> t : T \<Longrightarrow> e \<turnstile> t' : T"
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  by (induct set: rtranclp) (iprover intro: subject_reduction)+
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subsection {* Alternative induction rule for types *}
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lemma type_induct [induct type]:
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  assumes
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  "(\<And>T. (\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T1) \<Longrightarrow>
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    (\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T2) \<Longrightarrow> P T)"
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  shows "P T"
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proof (induct T)
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  case Atom
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  show ?case by (rule assms) simp_all
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next
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  case Fun
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  show ?case by (rule assms) (insert Fun, simp_all)
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qed
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end